The House of Representatives is only needed if there is a law on the books that says there can only be 9 Justices on the Supreme Court.
If there is no law, then yeah. Only need the Senate to change the amount of justices.
I would love it if there was a law about it as it would make it slightly harder for bad faith actors to expand the Court on the grounds of “The other side got to do it, so we get to do it too!” Aka Republicans.